Extremal Theorems in Random Discrete Structures

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Introduction New results

Extremal Theorems in Random Discrete Structures

J´ozsef Balogh

February 2013

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Recent trends in combinatorics

Classical extremal Theorems in combinatorics. Is the random sparse variant true?

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Recent trends in combinatorics

Classical extremal Theorems in combinatorics.

Is the random sparse variant true?

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Recent trends in combinatorics

Classical extremal Theorems in combinatorics.

Is the random sparse variant true?

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Recent trends in combinatorics

Example: Bounded degree Trees in graphs

Komlós, G. Sárközy, Szemerédi (1995)

Ifδ(Gn)≥(1 +o(1))n/2 thenTn ⊂Gn for every bounded degree tree.

P´osa; Friedman-Pippinger; Haxell; Alon-Krivelevich-Sudakov; Balogh, Csaba, Pei and Samotij (2010)

For every >0,d ifp > _n^d log¹ then w.h.p. G(n,p) contains every treeT with |T|<(1−)n, ∆(T)<d.

Balogh, Csaba, and Samotij (2011)

A.a.s. every subgraph of G(n,p) with minimum degree at least (1/2 +)np contains every bounded degree tree with (1−)n vertices, wherep >C()/n.

For what p will a.a.s. G(n,p) contain every bounded degree spanning tree? Johannsen, Krivelevich, Samotij (2012) p =n^−1/3+o(1) is sufficient.

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Recent trends in combinatorics

Example: Bounded degree Trees in graphs Komlós, G. Sárközy, Szemerédi (1995)

Ifδ(Gn)≥(1 +o(1))n/2 thenTn⊂Gn for every bounded degree tree.

P´osa; Friedman-Pippinger; Haxell; Alon-Krivelevich-Sudakov; Balogh, Csaba, Pei and Samotij (2010)

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Recent trends in combinatorics

P´osa; Friedman-Pippinger; Haxell; Alon-Krivelevich-Sudakov;

Balogh, Csaba, Pei and Samotij (2010)

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Recent trends in combinatorics

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Recent trends in combinatorics

For what p will a.a.s. G(n,p) contain every bounded degree spanning tree?

Johannsen, Krivelevich, Samotij (2012) p =n^−1/3+o(1) is sufficient.

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Recent trends in combinatorics

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Recent trends in combinatorics

Example: Triangle factors in graphs

Corr´adi, Hajnal (1963)

Ifδ(G3n)≥2n thenG3n contains a triangle-factor. Johansson, Kahn, Vu (2008)

Ifp n^−2/3(logn)^1/3 then w.h.p. G(3n,p) contains a triangle-factor.

Balogh, Lee, Samotij (2012)

For all γ >0, there exists C such that if p((logn)/n)^1/2, then a.a.s. everyH ⊂G(n;p) with δ(H)>(2/3 +γ)np contains a triangle packing that covers all but at most C/p² vertices.

Sudakov, Hao, Lee (2011) C/p² is best possible. What about larger cliques?

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Recent trends in combinatorics

Example: Triangle factors in graphs Corr´adi, Hajnal (1963)

Ifδ(G_3n)≥2n thenG_3n contains a triangle-factor.

Johansson, Kahn, Vu (2008)

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Recent trends in combinatorics

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Recent trends in combinatorics

Sudakov, Hao, Lee (2011) C/p² is best possible.

What about larger cliques?

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Recent trends in combinatorics

Sudakov, Hao, Lee (2011) C/p² is best possible.

What about larger cliques?

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Recent trends in combinatorics

Example:K_s-factors in graphs

Hajnal, Szemer´edi (1970)

Ifδ(G_sn)≥(s −1)n thenG_sn contains aK_s-factor. Johansson, Kahn, Vu (2008)

Ifp n−(s−1)/[s(s−1)](logn)^1/s then w.h.p. G(sn,p) contains a Ks-factor.

Balogh, Morris, Samotij (2012+)

Conlon, Gowers, Samotij, Schacht (2012++)

For every s ifp n^−2/(s+1), then a.a.s. every H⊂G(sn;p) with δ(H)>[s−1 +o(1)]np contains aKs-packing that covers all buto(n) vertices.

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Recent trends in combinatorics

Example:K_s-factors in graphs Hajnal, Szemer´edi (1970)

Ifδ(G_sn)≥(s −1)n thenG_sn contains aK_s-factor.

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Recent trends in combinatorics

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Recent trends in combinatorics

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Classical extremal Theorems in combinatorics

Example: Tur´an type Theorems

Tur´an (1941)

ex(n,K_k+1) =t_k(n) = (1−¹_k +o(1)) ⁿ₂ . Erd˝os, Frankl, R¨odl (1986)

There are 2(1+o(1))·ex(n,K_k+1) K_k+1-free graphs onn vertices. Kolaitis, Pr¨omel, Rotshchild (1987)

Almost all K_k₊₁-free graphs arek-partite. Related Sparse questions:

For what p is the following true?

ex(G(n,p),K_k+1) = (1− ¹_k +o(1))p ⁿ₂ . For what m=m(n) is the following true?

The number K_k+1-free graphs with medges is ^t^k^(n)+o(n_m ²⁾ . For what m=m(n) is the following true?

A.a. K_k+1-free graphs withm edges are (almost)k-partite.

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Classical extremal Theorems in combinatorics

Example: Tur´an type Theorems Tur´an (1941)

ex(n,K_k+1) =t_k(n) = (1−¹_k +o(1)) ⁿ₂ .

Erd˝os, Frankl, R¨odl (1986)

There are 2(1+o(1))·ex(n,K_k+1) K_k+1-free graphs onn vertices. Kolaitis, Pr¨omel, Rotshchild (1987)

Almost all K_k₊₁-free graphs arek-partite. Related Sparse questions:

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Classical extremal Theorems in combinatorics

There are 2(1+o(1))·ex(n,K_k+1) K_k+1-free graphs onn vertices.

Kolaitis, Pr¨omel, Rotshchild (1987) Almost all K_k₊₁-free graphs arek-partite. Related Sparse questions:

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Classical extremal Theorems in combinatorics

Kolaitis, Pr¨omel, Rotshchild (1987) Almost all K_k₊₁-free graphs arek-partite.

Classical extremal Theorems in combinatorics

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Classical extremal Theorems in combinatorics

ex(G(n,p),K_k+1) = (1− ¹_k +o(1))p ⁿ₂ .

For what m=m(n) is the following true?

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Classical extremal Theorems in combinatorics

The number K_k+1-free graphs withm edges is ^t^k^(n)+o(n_m ²⁾ .

For what m=m(n) is the following true?

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Classical extremal Theorems in combinatorics

The number K_k+1-free graphs withm edges is ^t^k^(n)+o(n_m ²⁾ . For what m=m(n) is the following true?

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Classical Extremal Theorems in Additive Combinatorics

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Classical extremal Theorems in combinatorics

Definition

We say thatA⊂[n] is (δ,k)-Szemer´edi, if everyB ⊂A with

|B|> δ|A|contains an arithmetic progression of lengthk. (k-AP).

TheoremSzemer´edi (1975)

For everyδ >0,k, for sufficiently large n, the set [n]is (δ,k)-Szemer´edi.

Endre Szemer´edi

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Classical extremal Theorems in combinatorics

Definition

For everyδ >0,k, for sufficiently large n, the set[n]is (δ,k)-Szemer´edi.

Endre Szemer´edi

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Classical extremal Theorems in combinatorics

Definition

For what p=p(δ,k) ap-random subset of [n] is w.h.p.

(δ,k)-Szemer´edi?

For a given m, how manym-subset of [n] does not contain an k-AP?

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Classical extremal Theorems in combinatorics

Definition

For what p=p(δ,k) ap-random subset of [n] is w.h.p.

(δ,k)-Szemer´edi?

For a given m, how manym-subset of [n] does not contain an k-AP?

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Generalizing Szemer´ edi’s theorem

Conjecture (Erd˝os) If R⊆N satisfiesP

n∈R 1

n =∞, then R contains arbitrarily long APs.

Theorem (Green–Tao [2004])

LetA⊆Pbe a subset of the primes whose upper density is positive, i.e.,

lim sup

n→∞

|A∩[n]|

|P∩[n]| >0 Then Acontains arbitrarily long APs.

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Generalizing Szemer´ edi’s theorem

Conjecture (Erd˝os, $5000) If R⊆N satisfiesP

n∈R 1

lim sup

n→∞

|A∩[n]|

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Generalizing Szemer´ edi’s theorem

Conjecture (Erd˝os, $5000) If R⊆N satisfiesP

n∈R 1

lim sup

n→∞

|A∩[n]|

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Generalizing Szemer´ edi’s theorem

Every subset of the primes with positive upper density contains arbitrarily long APs.

B. Green T. Tao

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Extremal problems – general framework

Framework Setting:

A finite setV and a k-uniform hypergraph H ⊆ P(V) on V.

Questions:

What is the size of the largest independent set in H? What are the largest independent sets in H?

How many independent sets does H have? What does a typical independent set look like? Questions: (Sparse random variants)

What is the size of the largest independent set in G^(v)(H,p)? How many independent sets of size mdoes Hhave?

What does a typical independent set of size mlook like?

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Extremal problems – general framework

Framework Setting:

A finite setV and a k-uniform hypergraph H ⊆ P(V) on V. Questions:

What is the size of the largest independent set in H? What are the largest independent sets in H?

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Extremal problems – general framework

Framework Setting:

What is the size of the largest independent set in H?

What are the largest independent sets in H? How many independent sets does H have? What does a typical independent set look like? Questions: (Sparse random variants)

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Extremal problems – general framework

Framework Setting:

What are the largest independent sets inH?

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Extremal problems – general framework

Framework Setting:

How many independent sets does Hhave?

What does a typical independent set look like? Questions: (Sparse random variants)

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Extremal problems – general framework

Framework Setting:

What does a typical independent set look like?

Questions: (Sparse random variants)

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Extremal problems – general framework

Framework Setting:

What is the size of the largest independent set in G^(v)(H,p)?

How many independent sets of size mdoes Hhave? What does a typical independent set of size mlook like?

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Extremal problems – general framework

Framework Setting:

How many independent sets of size mdoesH have?

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Extremal problems – general framework

Framework Setting:

How many independent sets of size mdoesH have?

What does a typical independent set of sizem look like?

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Extremal problems – general framework

Example: (Tur´an problem) V(H) =E(Kn),

E(H) = edge-sets of copies ofKk in Kn, H is ^k₂

-uniform,

Independent sets in H → K_k-free subgraphs inK_n.

Example: (Szemer´edi’s Theorem) V(H) ={1, . . . ,n},

E(H) =k-term APs in [n], H isk-uniform,

Independent sets in H → k-AP-free subsets of [n].

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Extremal problems – general framework

Example: (Tur´an problem) V(H) =E(Kn),

E(H) = edge-sets of copies ofKk in Kn, H is ^k₂

-uniform,

Independent sets in H → K_k-free subgraphs inK_n. Example: (Szemer´edi’s Theorem)

V(H) ={1, . . . ,n}, E(H) =k-term APs in [n], H isk-uniform,

Independent sets in H → k-AP-free subsets of [n].

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Sparse random analogues

Metatheorem (Conlon, Gowers; Schacht (2010+++))

’dense’ extremal result

+ =⇒ ’sparse’ extremal result.

supersaturation

Dr D. Conlon Sir W.T. Gowers Dr M. Schacht

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Sparse random analogues

supersaturation

Theorem (Conlon, Gowers; Schacht)

For every k and if p≥C(k)·n⁻^k+2² , then a.a.s., ex(G(n,p),Kk+1) =

1− 1

k +o(1) n 2

p.

For every k ≥3 andδ >0, if p≥C(k, δ)·n⁻^k−1¹ , then [n]p is w.h.p. (δ,k)-Szemer´edi.

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Sparse random analogues

supersaturation

For every k and if p≥C(k)·n⁻^k+2² , then a.a.s., ex(G(n,p),Kk+1) =

1− 1

k +o(1) n 2

p.

For every k ≥3 andδ >0, if p≥C(k, δ)·n⁻^k−1¹ , then [n]p is w.h.p. (δ,k)-Szemer´edi.

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Sparse random analogues

Metatheorem (Conlon, Gowers; Samotij)

’dense’ stability result

+ =⇒ ’sparse’ stability result removal lemma

Theorem (Conlon, Gowers)

For every k ≥2 and every δ >0, there exist C and ε >0 such that if p≥Cn⁻^k+2² , then a.a.s. every K_k+1-free subgraph of G(n,p) with at least(1−¹_k −ε) ⁿ₂

p edges may be made k-partite by removing at mostδn²p edges.

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Sparse analogues of counting problems

Definition

For a hypergraphH, let

I(H) := independent sets in H.

Definition

LetV be a (finite) set. A family F ⊆ P(V) isincreasing (anupset) ifA∈ F andB ⊇A implyB ∈ F.

Definition

LetHbe a k-uniform hypergraph,F ⊆ P(V(H)) an upset, and ε >0. We say thatHis (F, ε)-denseif for every A∈ F,

e(H[A])≥εe(H).

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Sparse analogues of counting problems

Definition

For a hypergraphH, let

I(H) := independent sets in H.

Definition

LetV be a (finite) set. A family F ⊆ P(V) isincreasing (anupset) ifA∈ F andB ⊇A implyB ∈ F.

Definition

LetHbe a k-uniform hypergraph,F ⊆ P(V(H)) an upset, and ε >0. We say thatHis (F, ε)-denseif for every A∈ F,

e(H[A])≥εe(H).

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Refined framework

Example

The following hypergraph is (F, ε)-dense:

V(H) = [n],

E(H) =k-term APs, F ={A⊆[n] :|A| ≥δn}.

Example

The following hypergraph is also (F, ε)-dense: V(H) =E(Kn),

E(H) = edge sets of copies ofKk+1, F = graphs with at least (1−1/k+ε) ⁿ₂

edges.

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Refined framework

Example

The following hypergraph is (F, ε)-dense:

V(H) = [n],

E(H) =k-term APs, F ={A⊆[n] :|A| ≥δn}.

Example

The following hypergraph is also (F, ε)-dense:

V(H) =E(Kn),

E(H) = edge sets of copies ofKk+1, F = graphs with at least (1−1/k+ε) ⁿ₂

edges.

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Main result

Theorem (Balogh, Morris, Samotij)

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Main result

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Main result

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Main result

For every k andε, there is m0 =m0(N) such that if H is an N-vertex k-uniform hypergraph which

is (F, ε)-dense for some upsetF ⊆ P(V(H)) and satisfies certain technical conditions, [bounds on degrees, co-degrees]

then there are

a familyS ⊆ ^V_≤m^(H)

0

and

functions f :S → F^c and g:I(H)→ S such that for every I ∈ I(H)

g(I)⊆I and I\g(I)⊆f(g(I)).

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Main result

then there are

0

and

functions f :S → F^c and g:I(H)→ S

such that for every I ∈ I(H)

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Main result

then there are

0

and

functions f :S → F^c and g:I(H)→ S such that for every I ∈ I(H)

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Corollaries

Implies most results of Conlon-Gowers, and Schacht:

Sparse Szemer´edi Theorem.

Sparse Tur´an Theorem.

Sparse Erd˝os- Stone Theorem (for balanced graphs).

Sparse stability theorem.

Proof is much shorter and simpler!

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Corollaries – Tur´ an problem

For every k andδ >0, if m≥C(k, δ)n^2−2/(k+2), then almost every K_k+1-free n-vertex graph with m edges can be made k-partite by removing from it at mostδm edges.

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Corollaries – Szemer´ edi Theorem

For every k ≥3 andδ >0, if m≥C(k, δ)n¹⁻^k−1¹ , then

#m-subsets of [n]with no k-term AP≤ δn

m

.

Corollary

For every k ≥3 and every δ⁰ >0, if p≥C(k, δ⁰)·n⁻^k−1¹ , then a.a.s.[n]_p is (δ⁰,k)-Szemer´edi.

Proof.

m:=δ⁰pn, δ:=δ⁰/e².

P [n]_p contains ak-term AP-free set of sizeδ⁰np

≤ ^δn_m

·p^m≤

eδnp δ⁰np

δ⁰np

=o(1).

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Corollaries – Szemer´ edi Theorem

m

.

Corollary

Proof.

m:=δ⁰pn, δ:=δ⁰/e².

≤ ^δn_m

·p^m≤

eδnp δ⁰np

δ⁰np

=o(1).

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Corollaries – Szemer´ edi Theorem

m

.

Corollary

Proof.

m:=δ⁰pn, δ:=δ⁰/e².

≤ ^δn_m

·p^m≤

eδnp δ⁰np

δ⁰np

=o(1).

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How to apply the meta-theorem?

m

.

Form the hypergraph Hwith V(H) = [n],E(H) ={k-AP}. Check if it satisfies co-degree conditions. (easily).

F :={A⊆[n] : |A| ≥δn}. (upset)

Averaging gives that every A∈ F contains at least cn² k-AP’s.

There are functions g:I(H)→ S andf :S → F^c. For given I independent set, S =g(I)⊂I ⊂f(S)∪S.

|S| is small, |f(S)|is small, so number of ways of choosingI is small.

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How to apply the meta-theorem?

m

.

Form the hypergraph Hwith V(H) = [n],E(H) ={k-AP}.

Check if it satisfies co-degree conditions. (easily). F :={A⊆[n] : |A| ≥δn}. (upset)

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How to apply the meta-theorem?

m

.

Check if it satisfies co-degree conditions. (easily).

F :={A⊆[n] : |A| ≥δn}. (upset)

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How to apply the meta-theorem?

m

.

F :={A⊆[n] : |A| ≥δn}. (upset)

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How to apply the meta-theorem?

m

.

F :={A⊆[n] : |A| ≥δn}. (upset)

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How to apply the meta-theorem?

m

.

F :={A⊆[n] : |A| ≥δn}. (upset)

There are functions g:I(H)→ S andf :S → F^c.

For given I independent set, S =g(I)⊂I ⊂f(S)∪S.

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How to apply the meta-theorem?

m

.

F :={A⊆[n] : |A| ≥δn}. (upset)

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How to apply the meta-theorem?

m

.

F :={A⊆[n] : |A| ≥δn}. (upset)

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On the structure of subsets of [n] with no k -term AP:

For every k ≥3, δ >0 there is a C such that for t = 2^Cn^1−1/k^logⁿ. there are F1, . . . ,Ft ⊂[n],

each of size at mostδn,

such that foreverysubset of [n]with no k-term AP there is an F_i containing it.

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The KLR-conjecture

Szemer´ediRegularity Lemma useful with Embedding Lemma:

(V1,V2),(V1,V3),(V2,V3) regular dense pairs then there is a triangle v₁v₂v₃ with v_i ∈V_i.

Sparse Regularity Lemma [Kohayakawa 97, R¨odl 97, Scott 11] Luczak: Embedding Lemma is false!

KLR-Conjecture, BMS-Theorem: Counterexamples are rare! Conlon, Gowers, Samotij, Schacht (2012++)

Counterexamples even for the counting lemma are rare in random graphs!

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The KLR-conjecture

Sparse Regularity Lemma [Kohayakawa 97, R¨odl 97, Scott 11] Luczak: Embedding Lemma is false!

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The KLR-conjecture

Sparse Regularity Lemma [Kohayakawa 97, R¨odl 97, Scott 11]

Luczak: Embedding Lemma is false!

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The KLR-conjecture

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The KLR-conjecture

KLR-Conjecture, BMS-Theorem: Counterexamples are rare!

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The KLR-conjecture

KLR-Conjecture, BMS-Theorem: Counterexamples are rare!

Conlon, Gowers, Samotij, Schacht(2012++)

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One proof idea for 2-uniform hypergraphs

Max degree ordering of H: v1 max degree vertex in H.

v2 max degree vertex in H −v1.. . .

Given I independent set, find S ⊂I ⊂f(S)∪S.

Let s1 =vi be first vertex of I. Removev1, . . . ,vi−1,N(s1). BuildS until |S|=Cpn, remaining alive vertices formf(S). density conditions imply that always many vertices are removed, i.e. |f(S)|is small.

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One proof idea for 2-uniform hypergraphs

Let s1 =vi be first vertex of I. Removev1, . . . ,vi−1,N(s1). BuildS until |S|=Cpn, remaining alive vertices formf(S). density conditions imply that always many vertices are removed, i.e. |f(S)|is small.

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One proof idea for 2-uniform hypergraphs

Let s1 =vi be first vertex of I. Remove v1, . . . ,vi−1,N(s1).

BuildS until |S|=Cpn, remaining alive vertices formf(S). density conditions imply that always many vertices are removed, i.e. |f(S)|is small.

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One proof idea for 2-uniform hypergraphs

BuildS until|S|=Cpn, remaining alive vertices formf(S).

density conditions imply that always many vertices are removed, i.e. |f(S)|is small.

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One proof idea for 2-uniform hypergraphs

BuildS until|S|=Cpn, remaining alive vertices formf(S).

density conditions imply that always many vertices are removed, i.e. |f(S)|is small.